35 research outputs found
A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2(Q), Q a non-singular quadric
AbstractWe study the small weight codewords of the functional code C2(Q), with Q a non-singular quadric in PG(N,q). We prove that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes. We also calculate the number of codewords having these small weights
Functional codes arising from quadric intersections with Hermitian varieties
AbstractWe investigate the functional code Ch(X) introduced by G. Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h=2. In [4], F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) [11] on the minimum distance of this code for a Hermitian variety X in PG(3,q2). In this paper, we will answer the question about the minimum distance in general dimension N, with N<O(q2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes
The small weight codewords of the functional codes associated to non-singular hermitian varieties
This article studies the small weight codewords of the functional code C (Herm) (X), with X a non-singular Hermitian variety of PG(N, q (2)). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q (2)) consisting of q + 1 hyperplanes through a common (N - 2)-dimensional space I , forming a Baer subline in the quotient space of I . The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C (2)(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729-1739, 2010), and C (2)(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27-35, 2010)
Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic
We determine the possible intersection sizes of a Hermitian surface with an irreducible quadric of sharing at least a
tangent plane at a common non-singular point when is even.Comment: 20 pages; extensively revised and corrected version. This paper
extends the results of arXiv:1307.8386 to the case q eve
Intersections of the Hermitian surface with irreducible quadrics in , odd
In , with odd, we determine the possible intersection sizes of
a Hermitian surface and an irreducible quadric
having the same tangent plane at a common point .Comment: 14 pages; clarified the case q=
Deformation classification of real non-singular cubic threefolds with a marked line
We prove that the space of pairs formed by a real non-singular cubic hypersurface with a real line has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface formed by real lines on . For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of characterizes completely the component
Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
We propose a new approach to the geometry of the four-qubit entanglement
classes depending on parameters. More precisely, we use invariant theory and
algebraic geometry to describe various stratifications of the Hilbert space by
SLOCC invariant algebraic varieties. The normal forms of the four-qubit
classification of Verstraete {\em et al.} are interpreted as dense subsets of
components of the dual variety of the set of separable states and an algorithm
based on the invariants/covariants of the four-qubit quantum states is proposed
to identify a state with a SLOCC equivalent normal form (up to qubits
permutation).Comment: 49 pages, 16 figure